Ind. Eng. Chem. Fundam. 1985, 2 4 , 121-128
121
Pinning of a Bed of Particles in a Vertical Channel by a Cross Flow of Gas Jean C. Glnestra and Roy Jackson’ UnlversHy of Houston, Houston, Texas 77004
For certain types of chemical reactor, whose catalyst decays reversibly and rather quickly, continuous operation is possible by use of a moving bed of catalyst which slides down through the reactor under gravity, while the reaction mixture is passed horizontally through the bed. There is then an upper limit on the flow rate of reactants, imposed by the mechanical phenomenon of “plnnlng”, in which the drag force exerted on the particles by the flowing reaction mixture presses the bed against the wall of the reactor so hard that friction at this wall prevents the bed from sliding downward. This paper presents a simple analysis of the mechanics of this situation and compares its predictions with observations in a rectangular channel.
Introduction For chemical reactors whose catalyst activity decays rapidly but reversibly, the traditional mode of operation uses two beds, which are switched alternately between reaction and regeneration. In the regeneration mode the stream of reactant gas is replaced by a stream containing substances which react with the catalyst contaminant. Perhaps the most extensive examples arise in the petroleum refining and petrochemicals industries, where the catalyst contaminant is “coke”,a highly carbonaceous solid deposit which can be removed by reaction with oxygen or steam. Regeneration frequently uses a flde gas consisting primarily of nitrogen, with a small proportion of oxygen, so that the temperatures achieved during regeneration do not damage the catalyst irreversibly. A more recent development is the continuously regenerated catalytic reactor, in which a moving bed of catalyst passes through, is regenerated after it leaves, and is then continually recycled back to the reactor. The most important example of this is the continuously regenerated catalytic reformer which uses a cross-flow moving bed, in which catalyst pellets move vertically downward through the reactor under gravity, while the reacting gas is blown horizontally across the bed through porous plates which form ita vertical retaining walls. Figure 1shows the most common configuration, using a bed of annular cross section, with the reactive mixture flowing radially, either inward or outward. An upper bound on the flow of reaction gas in such a system is imposed by the phenomenon of “pinning”, described by Bridgwater (1981). The drag force exerted by the flowing gas forces the bed against the downstream porous wall, thus increasing the frictional stress which opposes sliding of the material over this wall. If the gas flow is large enough, the resulting frictional force is sufficient to support the weight of the bed and downward motion ceases, at least in some region adjacent to the downstream wall. The bed is then said to be “pinned” by the gas flow, and the catalyst held immobile in this way becomes completely deactivated by coking, preventing continued operation of the reactor. The only accounts of this phenomenon in the open literature appear to be the brief discussion by Bridgwater (1981), mentioned above, and an aside by Trambouze (1979). The present paper describes a preliminary ex-
* Address correspondence to this author at the Department of Chemical Engineering, Princeton University, Princeton, NJ 08544. 0196-4~13/85/1024-0121$01.50/0
perimental and theoretical investigation of pinning in a system with simplified rectangular geometry. This exhibits the essentials of the phenomenon and is more accessible to observation than the more realistic cylindrical config uration. Theory of Cavity Growth We consider a bed of of particulate material moving slowly down a vertical channel of rectangular cross section, as shown in Figure 2. This channel contains a section in which a pair of opposing faces are permeable to air flow but do not permit the particles to pass through them. Air is blown horizontally through the bed between these two faces, entering by the one on the left (the upstream face) in Figure 2 and leaving through the one on the right (the downstream face). The horizontal section of the channel has dimensions D parallel to the direction of air flow and W perpendicular to this direction, while the total length of the aerated section is denoted by L. The drop in pressure of the air as it passes through the bed will be denoted by Ap, and for simplicity the streamlines of air flow will be assumed to be straight lines normal to the porous faces of the column. As A p is increased from zero the drag force exerted on the particles by the air causes the normal stress between the particles and the upstream porous face to decrease, with a corresponding increase in the normal stress at the downstream porous face. Denote these normal stresses by N1and N,, respectively. Since the particles are sliding downward in contact with the porous faces, corresponding frictional stresses Nl tan 6 and N2 tan 6 are mobilized and act vertically upward on the particle bed. As A p is increased, a value Apo w i l l be reached at which Nl is reduced to zero, and we would then expect a cavity to open between the particle bed and the upstream face of the channel. With further increase in A p the size of the cavity adjacent to the upstream face might be expected to increase, until it ultimately spans the full width of the channel. At this point, with pressure drop Apl, there can be no downflow of particles into the porous section, and the bed will be said to be completely pinned. We shall now develop a theoretical description of this process of cavity growth. Figure 3 shows the aerated section with a cavity adjacent to the upstream wall. A coordinate system is taken with the origin at the bottom of the downstream wall on the centerline of the channel and with the x axis directed vertically upward, the y axis pointing into the air flow, and the z axis perpendicular to the air flow. Conditions in the bed will be assumed uniform in the z direction and motion 0 1985 American Chemical Society
122
Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985 CATALYST
'i
Figure 1. Typical arrangement of a moving bed reactor with crossflow of reactant gas.
-I-"
2 - 4
XY
Qxx
Figure 3. Form of the particle bed within the aerated section, with pressure drop A p such that Ap, < Ap < Apl. The aerated section has porous walls of length L and width W .
+
and x bx, as indicated in Figure 3. The y component of this balance gives
while the corresponding x component is
Figure 2. Schematic arrangement of rectangular channel for experiments on pinning (all dimensions in mm).
is assumed to be slow enough to neglect inertial terms in the equations of motion. The stress tensor representing particle-particle contact forces in the bed will be defined in the compressive sense and its elements, evaluated on the downstream face y = 0, will be denoted by aij. All elements of stress must, of course, vanish on the free surface of the bed which forms the cavity wall, provided the particulate material is not cohesive. The dependence of the uij on x and y could be found by integrating the differential equations of force balance on the bed material, subject to initial conditions on the horizontal plane which spans the top of the aerated section and boundary conditions on the walls of the column. However, the initial conditions are subject to the usual uncertainty (Horne and Nedderman, 1976) regarding the stress field in a deep parallel-sided bin, so instead we shall adopt a simpler procedure, merely assuming that each element of the stress tensor varies linearly in the y direction between its value on the downstream face of the column and zero at the edge of the cavity. The necessary equations can then be obtained by writing a force balance on a thin slice of the bed between horizontal planes at x
where p is the bulk density of the bed, Y is its width in the direction of air flow, and E,, denote elements of the stress tensor evaluated on the downstream face Y = 0. The factors l/z in these equations arise from averaging over the width of the slab, with the assumption of linear variation of stress. To go further than this, some assumptions must be made regarding friction forces at the boundaries and the relation between elements of the stress tensor within the bed. As the bed moves down the shaded region in Figure 3, all or part of it must deform since the cross-sectional area changes. It will be assumed that the bed is deforming everywhere and that it behaves in continued deformation as a Coulomb material (Jackson, 1983), so that elements of XI, stress tensor are related by Mohr-Coulomb yield condition
[(Cy,- C,,)' + 4E,,211/2= (Exx + C,) sin 9 (3) where 9 is the angle of internal friction. Furthermore, since the particulate material slides over the vertical faces of the channel, the usual frictional relation between tangential and normal stresses gives
E,,
E,, C x z= E,,tan 8 = fCZz(say); E,, =
Cy,tan
6 = f C , (say);
=0
(4)
o
(5)
=
where 6 and 8 are the angles of friction of the sliding bed in contact with the downstream porous wall and with the impermeable walls at z = f W / 2 , respectively. Using (4) to eliminate Ex,from (3) then gives
Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985
E,,
= k C y y ; [(l- k)2
+ 4fL]1/2 = (1+ k ) sin 4
(6)
This has two roots in k , one smaller than unity and one larger. The appropriate one is chosen by recognizing that material moving down through the shaded region in Figure 3 is compressed in the horizontal direction and extended in the vertical direction, so we expect < E,. This is the case of “passive failure”, and the appropriate root of (6) is smaller than unity. It will be denoted by kl. It remains to find and we shall adopt the H m v o n Karman hypothesis (Haar and von Karman, 1909) that it matches one of the principal stresses in the (x,y) plane. In particular, for passive yield it will be equated with the larger of these. Since the projection of the stress tensor in the (x,y) plane is
e,,
e,,,
the principal stresses are the roots in
C/Cyy)(l- C / C J
E of the equation - f 2
=0
e,,is identified with el,the larger of these, so E,, = C1= E ,
(say)
(7)
where 1 is the larger root of (k1 - 1)(1 - 1) - f 2 = 0
(8)
The relations (3)-(8) now permit eq 1and 2 to be written entirely in terms of Cyy d (9) AP + Y2fZ(YCYY) =cyy
Eliminating the derivative between these then gives
cy,= ( k l A p - f p g V / ( kl - f
-
F)
and this may be substituted back into (9) to give a differential equation for the curve which defines the boundary separating the bed and the cavity, namely
dY _ dr
-
2(fQ - p g y
+ APflY/W)(ki - f - f f l Y / W )
(ki - f 2)(kiAP- 2 f ~ g V+ f 2 h p / W
(11) It is convenient to write this in dimensionless form, defining A = fAp/pgD; { = x / D ; q = Y / D (12) when it becomes
123
For a cavity to exist it is clearly necessary that dq/d{ = 1. Scrutiny of (13) immediately shows that dq/d{ = 0 when
> 0 at q
P
=
Po
= 1/( 1
+
7;)
and it is not difficult to show that the sign of dq/d{ changes from negative to positive as P increases through ?r0, for reasonable values of the remaining parameters. Then, for P = ro,the solution of (13) is the vertical line q = 1 coincident with the upstream face of the aerated section, and this corresponds physically to incipient cavity formation. r 0will, therefore, be referred to as the dimensionless pressure drop at cavity initiation. As P is increased progressively above r 0the cavity wall will move to smaller values of q, until a value P = r1is reached, at which the cavity wall profile intersects the downstream face of the aerated section at its lower extremity; in other words, the solution of (13) passes through the origin of the (3;s)plane. At this condition there can be no further downflow of particulate material through the aerated section, so the bed below would be expected to retreat down the column, leaving the upper part of the bed supported above a free surface (the solution of (13)) which spans the width of the aerated section from the top of the upstream face to the bottom of the downstream face. The bed will then be said to be completely pinned, and r1will be called the dimensionless pressure drop for complete pinning. This simple theory is not able to predict whether, or where, downward motion of particles may persist within so observations of the aerated section for ro< P < r1, pinning must be viewed in relation to the location of the calculated interval [ro, 7r1]. The relation between ?rl and L I D at complete pinning is obtained by integrating (13) between the points ( L I D , 1)and (0, 0 ) ,which gives 1
2L =
[
( k , - f z)(
(Pl
-
- 2fq)
+ r $fg W q)(kl
f 2f1$q2]
dq
- f 2 - f fDl F q )
(15) This integral diverges logarithmically at q = 1when ?rl = r0,as given by (14). Physically, this means that cavity initiation and complete pinning occur at the same value of a when the aerated section is unbounded in length. As L I D decreases, the value of r1 increakes monotonically, and it is easy to show from (15) that r1 m when
-
L k1-f2 In D-- - D 2flW
For given values of the parameters, f , f , Itl, I , D l W , and a selected value of A, eq 13 may be integrated to give the profile of the cavity wall. Since this starta from the upper edge of the upstream porous face, the integration may be started from the point ({, q) = ( L I D , 1 ) and continued through decreasing values of {to { = 0 at the bottom of the aerated section. Here the cavity ends and the full column width is, once again, expected to be occupied by the bed.
For values of L I D smaller than (16) complete pinning is not possible, however large the applied pressure drop. Consequently, the curve relating r1to D I L generated by eq 15 approaches a vertical asymptote. As the cavity develops with increasing Ap, there will be a progressive increase and redistribution of the gas flow through the aerated section, with more passing through the lower section where the cavity width is greatest. This is significant from the point of view of reactor performance, and it can be calculated without difficulty. However, for the mechanics of pinning, only the value of Ap is important.
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Ind. Eng. Chem. Fundam.. Vol. 24, No. 2, 1985
'1
02
,
I
,
02
04
06
ASYMPTOTE ,
O F r
,
~-, ,
,
,
0
0
08
IO
12
I4
16
D/,
Figure 4. Predicted curves relating xo and r1 to DIL: D / W = 0.5, 6 = 17O, 8 = 15O, 4 = 32.4O. Shaded region is domain of cavity growth.
1-
Figure 6. Cavity wall profile at complete pinning for aerated section of unbounded length: x1 = ro = 0.6667,D / W = 0.5,6 = 17O, 8 = 15', = 32.4O. C#J
AIR FLOW
Figure 5. Successive stages of cavity growth r0 = 0.6667, r l = 1.2,
LID = 0.865, DIW = 0.5, 6 = 17O, 8 = 15O, and 4 = 32.4O.
The theoretical predictions will be illustrated by specific + x l a t i o n s for the following parameter values: f = 0.306, f = 0.268, k , = 0.488, 1 = 1.143, and D/W = 0.5. These are appropriate for the system described below, which was used in our experimental work. Figure 4 then shows curves of a. and r1vs. DIL. The shaded region between these curves is the range of cavity growth. For the particular value L I D = 0.865, Figure 5 shows cavity profiles, calculated by integrating eq 13, for successively increasing values of a between a0 and al. The profiles are slightly convex downward and this convexity becomes much more pronounced as LID is increased, as seen in Figure 6, which shows the cavity profile at complete pinning for an aerated section of unbounded length.
Limitations Imposed by the Overburden The above theory of cavity growth confines attention to the material moving down the aerated section. In this region it follows from the expression below eq 10, and the assumption that the stress components vary linearly with y, that r
In particular, at the top of the aerated section where y = D
and it has been assumed implicitly that there exists some mechanism by which these stresses are applied across a
horizontal plane where the granular material enters the porous section. Physically, of course, the mechanism in question is the load generated by the overburden of particulate material in the vertical channel above the porous section. But it is well known (Cowin, 1977) that the stress due to such an overburden is bounded, no matter how great the depth, while according to eq 17 cXxincreases without bound as Ap increases. It therefore follows that some part of the curve of alvs. DIL in Figure 4, where it rises toward its asymptote, cannot be realized in the physical system considered. The limitations impceed by the nature of the overburden can properly be identified only by solving the complete partial differential equations of force balance throughout the channel, including the porous section. We shall not attempt anything so ambitious, but will try to get some feel for this aspect of the problem within the framework of cross-sectionally averaged force balances. The linear variation of stress with y in eq 17 is an arbitrary assumption, so the detailed form of the stress field given by eq 17, and the equation above it, must not be taken too seriously. Instead we shall consider only the total normal force exerted on the material in the porous section across the horizontal plane at the top of this section, namely
Using (17), this becomes
which must be matched by the total normal force exerted on this plane by the overburden. The theory of stresses in deep bins with vertical walls is well-known (Cowin, 1977). The orientation of the principal axes of stress in this system is ambiguous and depends on the history of the material, but it is commonly , k , and assumed that uzx = Kay,, where K E [ k 1 , k 2 ]and k 2 are the smaller and larger roots of eq 6, respectively. K = k l corresponds to the situation in which the material
Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985
125
is in passive failure throughout the bin, with the major principal stress axis horizontal on the central vertical line. On the other hand, K = k2 corresponds t o active failure throughout the bin, with the major principal stress axis vertical on the central line. At great depth below the free upper surface of the material, the stresses become independent of x , and in this asymptotic region a force balance on a thin horizontal slice of the material gives pgDW = 2fWuyy + 27Du,, in which the left-hand side represents the weight of the material, while the first and second terms on the righthand side represent frictional forces at the walls perpendicular to y and to z, respectively. But a,, = Kuyyand u,, = layy,so the above reduces to
1
I
(WLI,~
I
I (OIL),
D/L
Figure 7. Sketch of
Consequently, the total normal force exerted on a horizontal plane is given by
which represents the total normal force exerted by a deep overburden, however, great its depth. Equating F, and F l , given by eq 18 and 19, respectively, then balances the total normal force provided by the overburden with the total normal force needed to drive the material down into the porous section. Provided the resulting equation can be satisfied by a value of K in the interval [k,,k2],the overburden, is capable of generating the required force, and the motion in question is not forbidden by these simple force balance considerations. Consider the extreme values of K. When K = kl equating Fx and F,' gives AP=
PgD /
(1
7
-
\
+ 1;;)
or, in terms of the dimensionless variables
which coincides with the value given by eq 14, and therefore corresponds to incipient cavity formation. Thus, at the value of Ap which initiates a cavity at the upstream wall of the porous section, the overburden can provide the required normal force while moving down the channel in passive failure. When K = k2,on the other hand, equating F, and F,' gives a = a2 =
'kt2,
[ E( To
-
1-
a)]
(20)
At this value of a,the total normal force required to drive the material into the porous section matches the largest normal force which the overburden can possibly exert. If a is increased beyond this value a solution of the form found here, with orderly flow from the channel above into a region in the porous section bounded to the left by a gas cavity is no longer possible. The pinning diagram shown in Figure 4 must, therefore, be supplemented by a horizontal line determined by eq
?ro
line, x1 curve, and
1r2
line.
20, to give a complete diagram of the form sketched in Figure 7. In this diagram the alcurve has an asymptote at D I L = (DIL),, while the alcurve and the a2 line intersect at a smaller value, D I L = (DIL),. When D / L < (D/L),, the sequence of events when a increases is as follows: when a reaches the value r1,a cavity opens on the upstream wall of the porous section; it grows as a increases and meets the downstream wall when a = al. Thus there is complete pinning at a = xl. When D I L > (D/L),, on the other hand, a reaches the value azbefore the gas cavity has grown to span the full width of the channel. For values of a larger than a2the overburden can no longer force material down into the porous section, so the mode of flow analyzed here breaks down, even though complete pinning, in the sense defined above, has not been achieved. The present theory does not predict what will happen in these circumstances. For the parameter values used in constructing Figure 4, a2 = 3.228, so the a2 line is off the diagram shown.
Experimental Apparatus Figure 2 shows the arrangement of the apparatus used to study pinning in a bed of rectangular cross section. All dimensions indicated are in millimeters. Particles are fed to the test column from a hopper with capacity sufficient to permit about 15 min of steady operation. They then move down an entry section without air flow, whose purpose is to ensure that a fully developed pattern of motion is established before the cross flow of air is encountered. This is followed by the aerated section, whose front and back faces are of 0.02 in. thick sheet steel, perforated with a closely spaced pattern of holes whose dimensions me such that the particles cannot pass through them. Air is supplied to the upstream face via a windbox, which contains a perforated grid to disperse the jet of air from the inlet pipe. A tap on the inlet pipe permits the air pressure to be measured by means of a manometer. Since the downstream face discharges directly to atmosphere, the manometer reading gives a direct measure of the pressure drop across the system. The pressure drop across the bed itself is found by deducting the pressure drop measured in a blank calibration using the empty column. The aerated section is separated from a particle flow control valve at the foot of the column by an exit section, whose lower end is in the form of a small hopper to feed the particles into the flow control valve. It is important that the air should traverse the aerated section in pure cross flow. A significant leakage of air up
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Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985
air flows are needed for pinning. However, the cleanness and resistance to attrition of the glass beads are conducive to achieving reproducible results in experimental runs. The angles of friction appearing in the theory are measured using a Jenike tester, and have the following values: 6 = 1 7 O , 8 = 1 5 O , and 6 = 32.4'. In addition to experiments in a system with the dimensions shown in Figure 2, some work was done with the effective width D of the column reduced to 25.4 mm by inserts designed to fit within the casing of the column. In this way aerated sections with a larger L I D ratio could be investigated without having to increase L to the point of limitation by the capacity of the laboratory air supply.
FDR CONE
Figure 8. Arrangement of the particle flow control valve.
the entry section and out through the feed hopper, or down the column and out through the flow control valve, would clearly invalidate the results because of axial drag forces exerted on the particles. Accordingly, the apparatus is operated as a sealed system which air may enter and leave only through the perforated faces of the aerated section. The feed hopper is closed by a gasketed lid, and the particles are discharged into a container which is also sealed during an experimental run. Of course, in this confiiation there is necessarily an upflow of air displaced from the receiver by the descending particles. However, since the downward velocity of the particles is typically controlled at about 1 mm/s, the mechanical effect of this air flow is negligible. To achieve a steady flow of particles at this very small velocity, careful design of the particle flow control valve was necessary, and the arrangement ultimately used is shown in Figure 8. It consists of two steel cylinders, threaded to screw into each other, as shown. The inner cylinder is attached to the small hopper at the bottom of the column by an extensible hose, while the outer cylinder passes into the particle receiver through a bulkhead fitting. The cylinder can be rotated relative to this fitting, but an O-ring seals against air leakage. A cone, linked by rods to the bottom end of the outer cylinder, is centered axially below the end of the inner cylinder. When the outer cylinder is rotated the inner cylinder moves up or down, thus changing the width of the gap through which the particles must escape, between the cone and the bottom of the inner cylinder. The fine pitch of the thread on the cylinders permits close regulation of the particle flow rate, and this can be controlled stably down to very small values. Mass flow rates were typically in the range 3-30 kg/h. Air is fed to the windbox of the aerated section from the laboratory compressed air supply via a refrigerated dryer and filters to remove particles and suspended oil droplets. The pressure is then reduced by a regulator and the flow controlled by a hand valve and measured by rotameters. The particles used are glass beads with diameters in the range 0.85-1.25 mm. These are large enough to be held in the column by the perforated walls of the aerated section, but small enough in relation to the column dimensions that our continuum theory is credibly applicable. The bulk density of the bed is 1650 kg/m3, larger than that of a bed of reforming catalyst pellets, so relatively large
Experimental Results and Comparison with Theory To conduct an experimental run the particle flow control valve was first adjusted so that the bed moved downward steadily with a velocity of about 1 mm/s; then the cross flow of air was increased progressively until pinning occurred. The sequence of events in the aerated section was meanwhile observed, both through the transparent side wall of the column and, using a magnifying glass, through the holes in the porous downstream face of the column. Since this face is the primary seat of pinning, the latter observation was valuable for determining when particles in contact with the face came to rest. The sequence of events observed was qualitatively similar to that predicted theoretically and described above. First, a narrow cavity opened at the upstream face of the aerated section. At a certain pressure drop Apo this began to grow with increasing Ap until it spanned the full width of the channel, in the manner associated with complete pinning by the theory. The corresponding value of Ap is denoted by Ap,. For aspect ratios LID greater than 2, the difference between Ap, and Ap, was very small, and within this narrow range of pressure drops both the particles in contact with the downstream face and those in contact with the side walls of the channel were seen to come to rest. Because Ap, differed so little from Apo, it was not possible to discern whether the particles came to rest progressively with increasing Ap, starting from the downstream face, or whether they were all arrested simultaneously. Though the theory predicts that flow to the lower part of the column ceases at complete pinning, this was not always observed experimentally. The interface between the cavity and the bed in the aerated section has a tendency to instability, with particles crumbling away from it and raining down through the cavity. These provide a feed to the lower part of the column quite independent of any sliding of the particle bed on the walls of the aerated section. This phenomenon of particle flow by erosion from the cavity wall is not encompassed by the theory, and indeed it was not very reproducible in practice. In our earlier experiments this mechanism prevented any complete shut-off of flow down the column, but in later work complete pinning as predicted theoretically was observed, with a stable cavity wall and no flow into the lower part of the column. It is possible that this change in behavior was associated with contamination by a small amount of alumina dust, which may have occurred during storage of the particles. We believe that flow resulting from erosion of the cavity wall is a secondary (though by no means negligible) effect, while the formation of the cavity and the immobilization of particles within the bed are the phenomena which should properly be examined in relation to the theoretical predictions. Figure 9 shows tracings of the cavity wall profile from photographs of pinning in an aerated section with D = 76.2
Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985 127
0.41
Figure 9. Cavity wall proviles: (a) for ?ro < ?r < ?rl, and (b) for complete pinning with ?r = ?rl: traced from photographs,LID = 1.25, D /W = 0.5,6 = 17", 8 = 15O, 6 = 32.4O. Note upper surface of bed below aerated region in (b). This is moving downward.
mm and L I D = 1.25. In Figure 9a the cavity extends part way across the column, and its shape is similar to that predicted theoretically, except that the lower right corner is rounded, rather than sharp. This appears to be the result of a buildup of material which has fallen through the cavity after eroding from the free surface of the bed. In Figure 9b the pressure drop for complete pinning has been reached, the cavity spans the column, and there is no downflow of particulate material into the column below the aerated section. Indeed, the photograph shows the bed retreating downward below the pinned material, and the position of the upper surface of this bed at the time of the photograph is indicated on the diagram. Note that a rather small value of L I D was used in this example in order that the difference between Apo and Apl should be large enough to permit cavities of intermediate size, such as that shown in Figure 9a, to be stabilized and photographed. At the smallest aspect ratio investigated, namely L I D = 0.629, the cavity did not span the fullwidth of the column even with the largest value of Ap attainable. Because of erosion of particles from the surface of the cavity it is difficult to determine the value of Apl with precision. As an arbitrary measure of the pressure drop for pinning we therefore adopted that value of A p for which the particles in contact with the downstream face of the column are brought to rest. This could be recognized without difficulty by observing the particles with a magnifying glass through the holes in this porous face. In Figure 10 the results of these observations are plotted as the dimensionless pinning pressure drop, a,vs the aspect ratio D I L of the aerated section. There are two series of points, one taken in the full width column ( D = 76.2 mm) and one with the width reduced to D = 25.4 mm by inserts. Vertical bars indicate the rather large uncertainties in the values of a. These are associated with scatter in repeated measurements of Ap, possibly due to variations in the stress field at the top of the aerated section due to the overburden of descending particles. Superimposed on the experimental results are theoretical curves relating r1to D I L , and there are two of these, corresponding to the different values of D l W associated with the two values of D used in the experiments. Though the experimental points and the theoretical predictions nowhere differ by more than about 20% (except for the point at the highest value of D I L , where we have already commented that complete pinning could not be achieved), it is difficult to draw any conclusion regarding the success of the theory, since the observations are not made at complete pinning, as defined theoretically. However, it
0
0
I
EXPERIMENTAL POINTS
0.2
0.4
0.6
0.8
1.0
1.2
:",r:,'m",m 1.4
1.6
D/ L
Figure 10. Comparison of observed values of ?r for which particles in contact with downstream face are brought to rest, with theoretically predicted curves of ?rl vs. DIL: W = 152.4 mm, 6 = 17", 8 = 15O, and $J = 32.4'. Points are shown for two different values of D , as indicated.
Figure 11. Comparison of theoretical and observed cavity wall profiles at complete pinning: LID = 2.5, D /W = 0.5, d = 17", 8 = 15", and = 32.4O.
is clear that the difference between the sets of experimental points,taken for the two different values of D , is not nearly so large as the theory predicts. This suggests that friction at the side walls of the column may play a less important role than anticipated, and consequently that the Haar-von Karman hypothesis may not be an appropriate way to determine the normal stress uzz. If uzzwere instead set equal to the mean of the principal stresses in the (x,y) plane, which would be appropriate for a three-dimensional yield surface of circular cross section, the predicted values of a for the two values of D / W would be closer together, but both would be larger. Of course, if we set 6 = 0, the effect of friction at the side walls would be eliminated altogether and the predicted values of a would be independent of D / W. Furthermore, the curve of ?r vs. D I L would then start from a = 1 where D / L = 0. Finally, it is possible to compare the theoretically predicted and experimentally observed shapes of the cavitybed interface. Figure 11 shows the shape of the observed cavity wall at complete pinning, with D = 76.2 mm and L I D = 2.5, traced from a photograph taken through the transparent side wall of the column. Also shown is the
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Ind. Eng. Chern. Fundarn., Vol. 24, No. 2, 1985
profile predicted by integrating eq 13 for these conditions. The agreement is generally good, though the observed cavity is wider than predicted in the upper part of the aerated section. This might be expected as a result of the tendency for particles to crumble away from the free surface of the bed. However, it may also be due, in part, to the simplifying assumption that the air streamlines are horizontal. In fact, there will be a fringe effect associated with curvature of the streamlines into the bed above the aerated section.
Conclusions The rather simple theory presented here is based on the assumption that the particle bed behaves as a Coulomb material, everywhere in the condition of passive yield, and it uses only averaged force balance equations associated with elements which span the width of the bed. Nevertheless, it appears to give an account of the pinning phenomenon which is qualitatively correct and to provide quantitative estimates of pinning conditions which are in error by no more than about 20%. Since the ratio WID for the experimental column is not very large, the theoretical predictions are influenced significantly by the treatment of stresses normal to the plane of motion. This requires an assumption regarding the shape of the full three-dimensional yield surface in principal stress space, or some equivalent assumption such as the Haar-von Karman hypothesis invoked here. There is very little independent evidence bearing on the crosssectional shape of a conical Mohr-Coulomb yield surface, so this part of the theory must be regarded as tentative. It would, therefore, be valuable to perform experiments in columns with much larger values of WID,so that this source of uncertainty in the theory is less significant. However, the demands imposed on the air supply then increase, and our work was limited by the capacity of the available air supply. The plane geometry treated here is not realistic. The simplest practical devices consist of coaxial cylinders with the bed moving down the annular gap between them. In this case the treatment of stresses normal to the plane of yield cannot be avoided. It is hoped to report work on this geometry in a future publication. Acknowledgment The experiments were carried out in the laboratories of the Department of Chemical Engineering, University of Houston. ,J.C.G. wishes to acknowledge support by the
University and by the National Science Foundation, Particulate and Multiphase Processes Program. Part of the work was done while R.J. was in residence as a Fairchild Scholar at California Institute of Technology. Thanks are also due to Dan Luss, who first brought the problem to the authors' attention, and to Andrew Scott, who later asked the right question. Nomenclature D = width of channel in direction of air flow f = tan 6 f = tan 6 g = specific gravitational force k = UXJU,,) 1 = %z/Uy
L = length of aerated section in direction of particle motion N , , N 2 = normal stresses exerted by particles on upstream and downstream faces of channel, respectively p = air pressure W = width of channel normal to direction of air flow x , y , z = Cartesian coordinates, with y directed vertically upward and x in direction opposite to air flow Y = y coordinate of the surface of the air cavity in the aerated section Greek L e t t e r s 6 = angle of friction between particle bed and porous wall of
aerated section
S = angle of friction between particle bed and Plexiglas walls of channel { = dimensionless coordinate, x / D 7 = dimensionless coordinate, y / D a = dimensionless air pressure drop, fAp/pgD a. = value of a at initiation of cavity growth a, = value of a at complete pinning p = bulk density of bed of particles uV = elements of particle phase stress tensor,
defined in the compressive sense Z,] = value of u, at the downstream face y = 0 = larger of the principal stresses in the (r,y) plane 4 = angle of internal friction of bed of particulate material Literature Cited Bridgwater, J. Inaugural Lecture, University of Birmingham, 1981. Cowin, S.C. J. Appl. Mech. 1977, 4 4 , 409. Haar, A.; von Karman, T. Nachr. Wlss . Gottingen, Math .-Phys . Kksse, 1909, 204. Horne, I?.M.;Nedderman, R. M. Powder Techno/. 1978, 14, 93. Jackson, R. I n "Theory of Dlspersed Multiphase Flow", Meyer, E., Ed.: Academic Press: New York, 1983: p 291. Trambouze, P. Chem. Eng. 1979, 86(19), 122.
Received for review January 30, 1984 Accepted August 2 , 1984
